Newspace parameters
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.cg (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.35983195036\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
145.1 | −0.707107 | + | 0.707107i | −0.866025 | − | 0.500000i | − | 1.00000i | −4.23240 | + | 1.13407i | 0.965926 | − | 0.258819i | 1.34853 | + | 2.27629i | 0.707107 | + | 0.707107i | 0.500000 | + | 0.866025i | 2.19085 | − | 3.79467i | |
145.2 | −0.707107 | + | 0.707107i | −0.866025 | − | 0.500000i | − | 1.00000i | −1.30432 | + | 0.349490i | 0.965926 | − | 0.258819i | −1.83449 | − | 1.90648i | 0.707107 | + | 0.707107i | 0.500000 | + | 0.866025i | 0.675164 | − | 1.16942i | |
145.3 | −0.707107 | + | 0.707107i | −0.866025 | − | 0.500000i | − | 1.00000i | −0.981662 | + | 0.263036i | 0.965926 | − | 0.258819i | 2.09303 | − | 1.61840i | 0.707107 | + | 0.707107i | 0.500000 | + | 0.866025i | 0.508146 | − | 0.880134i | |
145.4 | −0.707107 | + | 0.707107i | −0.866025 | − | 0.500000i | − | 1.00000i | 3.01304 | − | 0.807342i | 0.965926 | − | 0.258819i | 2.56888 | + | 0.633142i | 0.707107 | + | 0.707107i | 0.500000 | + | 0.866025i | −1.55967 | + | 2.70142i | |
145.5 | −0.707107 | + | 0.707107i | −0.866025 | − | 0.500000i | − | 1.00000i | 3.50534 | − | 0.939253i | 0.965926 | − | 0.258819i | −2.61012 | − | 0.432735i | 0.707107 | + | 0.707107i | 0.500000 | + | 0.866025i | −1.81450 | + | 3.14280i | |
145.6 | 0.707107 | − | 0.707107i | −0.866025 | − | 0.500000i | − | 1.00000i | −2.31481 | + | 0.620250i | −0.965926 | + | 0.258819i | −2.57277 | + | 0.617120i | −0.707107 | − | 0.707107i | 0.500000 | + | 0.866025i | −1.19823 | + | 2.07540i | |
145.7 | 0.707107 | − | 0.707107i | −0.866025 | − | 0.500000i | − | 1.00000i | −1.42776 | + | 0.382567i | −0.965926 | + | 0.258819i | 0.349274 | + | 2.62260i | −0.707107 | − | 0.707107i | 0.500000 | + | 0.866025i | −0.739062 | + | 1.28009i | |
145.8 | 0.707107 | − | 0.707107i | −0.866025 | − | 0.500000i | − | 1.00000i | 0.499350 | − | 0.133800i | −0.965926 | + | 0.258819i | 0.430450 | + | 2.61050i | −0.707107 | − | 0.707107i | 0.500000 | + | 0.866025i | 0.258483 | − | 0.447705i | |
145.9 | 0.707107 | − | 0.707107i | −0.866025 | − | 0.500000i | − | 1.00000i | 0.589284 | − | 0.157898i | −0.965926 | + | 0.258819i | −1.91156 | − | 1.82919i | −0.707107 | − | 0.707107i | 0.500000 | + | 0.866025i | 0.305036 | − | 0.528338i | |
145.10 | 0.707107 | − | 0.707107i | −0.866025 | − | 0.500000i | − | 1.00000i | 2.65393 | − | 0.711118i | −0.965926 | + | 0.258819i | 1.40673 | − | 2.24078i | −0.707107 | − | 0.707107i | 0.500000 | + | 0.866025i | 1.37378 | − | 2.37945i | |
241.1 | −0.707107 | − | 0.707107i | −0.866025 | + | 0.500000i | 1.00000i | −4.23240 | − | 1.13407i | 0.965926 | + | 0.258819i | 1.34853 | − | 2.27629i | 0.707107 | − | 0.707107i | 0.500000 | − | 0.866025i | 2.19085 | + | 3.79467i | ||
241.2 | −0.707107 | − | 0.707107i | −0.866025 | + | 0.500000i | 1.00000i | −1.30432 | − | 0.349490i | 0.965926 | + | 0.258819i | −1.83449 | + | 1.90648i | 0.707107 | − | 0.707107i | 0.500000 | − | 0.866025i | 0.675164 | + | 1.16942i | ||
241.3 | −0.707107 | − | 0.707107i | −0.866025 | + | 0.500000i | 1.00000i | −0.981662 | − | 0.263036i | 0.965926 | + | 0.258819i | 2.09303 | + | 1.61840i | 0.707107 | − | 0.707107i | 0.500000 | − | 0.866025i | 0.508146 | + | 0.880134i | ||
241.4 | −0.707107 | − | 0.707107i | −0.866025 | + | 0.500000i | 1.00000i | 3.01304 | + | 0.807342i | 0.965926 | + | 0.258819i | 2.56888 | − | 0.633142i | 0.707107 | − | 0.707107i | 0.500000 | − | 0.866025i | −1.55967 | − | 2.70142i | ||
241.5 | −0.707107 | − | 0.707107i | −0.866025 | + | 0.500000i | 1.00000i | 3.50534 | + | 0.939253i | 0.965926 | + | 0.258819i | −2.61012 | + | 0.432735i | 0.707107 | − | 0.707107i | 0.500000 | − | 0.866025i | −1.81450 | − | 3.14280i | ||
241.6 | 0.707107 | + | 0.707107i | −0.866025 | + | 0.500000i | 1.00000i | −2.31481 | − | 0.620250i | −0.965926 | − | 0.258819i | −2.57277 | − | 0.617120i | −0.707107 | + | 0.707107i | 0.500000 | − | 0.866025i | −1.19823 | − | 2.07540i | ||
241.7 | 0.707107 | + | 0.707107i | −0.866025 | + | 0.500000i | 1.00000i | −1.42776 | − | 0.382567i | −0.965926 | − | 0.258819i | 0.349274 | − | 2.62260i | −0.707107 | + | 0.707107i | 0.500000 | − | 0.866025i | −0.739062 | − | 1.28009i | ||
241.8 | 0.707107 | + | 0.707107i | −0.866025 | + | 0.500000i | 1.00000i | 0.499350 | + | 0.133800i | −0.965926 | − | 0.258819i | 0.430450 | − | 2.61050i | −0.707107 | + | 0.707107i | 0.500000 | − | 0.866025i | 0.258483 | + | 0.447705i | ||
241.9 | 0.707107 | + | 0.707107i | −0.866025 | + | 0.500000i | 1.00000i | 0.589284 | + | 0.157898i | −0.965926 | − | 0.258819i | −1.91156 | + | 1.82919i | −0.707107 | + | 0.707107i | 0.500000 | − | 0.866025i | 0.305036 | + | 0.528338i | ||
241.10 | 0.707107 | + | 0.707107i | −0.866025 | + | 0.500000i | 1.00000i | 2.65393 | + | 0.711118i | −0.965926 | − | 0.258819i | 1.40673 | + | 2.24078i | −0.707107 | + | 0.707107i | 0.500000 | − | 0.866025i | 1.37378 | + | 2.37945i | ||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.ba | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 546.2.cg.b | yes | 40 |
7.d | odd | 6 | 1 | 546.2.by.b | ✓ | 40 | |
13.f | odd | 12 | 1 | 546.2.by.b | ✓ | 40 | |
91.ba | even | 12 | 1 | inner | 546.2.cg.b | yes | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.by.b | ✓ | 40 | 7.d | odd | 6 | 1 | |
546.2.by.b | ✓ | 40 | 13.f | odd | 12 | 1 | |
546.2.cg.b | yes | 40 | 1.a | even | 1 | 1 | trivial |
546.2.cg.b | yes | 40 | 91.ba | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{40} + 12 T_{5}^{38} + 12 T_{5}^{37} - 348 T_{5}^{36} + 128 T_{5}^{35} - 4680 T_{5}^{34} - 5592 T_{5}^{33} + 108147 T_{5}^{32} - 132044 T_{5}^{31} + 1446062 T_{5}^{30} - 752596 T_{5}^{29} - 6554768 T_{5}^{28} + \cdots + 14841086976 \)
acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).